Geometry 13.1 The Distance Formula

A(-5, 4). . B(-1, 4) Example 1 Find the distance between the two points. a. A(–5, 4) and B(–1, 4) b. C(2, –5) and D(2, 7) | -5 – (-1) | | 7 – (-5) | 4 units 12 units As you can see, if two points share an x coordinate or a y coordinate, the distance between the two points can be found by finding the distance (the absolute value of the difference) between the other coordinates. What if two points do not lie on a horizontal or vertical line?

. (1, -4) Why? Let’s try an example to find out! (-3, 4). Example 2 When two points do not lie on a horizontal or vertical line, you can find the distance between points by using the distance formula which is derived from the Pythagorean Theorem. The distance d between points and is: Why? Let’s try an example to find out! (-3, 4). Example 2 Find the distance between (–3, 4) and (1, –4). 4 4√5 . (1, -4) 8 Pythagorean Theorem!

Use the distance formula to find the distance between the two points. 4. (4, 2) and (1,-1) Answer: 3√2

Example 3 Given points A(1, 1), B(–3, –11), and C(4, 10), find AB, AC, and BC. Are A, B, and C collinear? If so, which point lies between the other two? AB = √(1+3)2 + (1+11)2 BC = √(-3 – 4)2 + (-11 - 10)2 AC = √(1 – 4)2 + (1 – 10)2 = √16 + 144 = √49 + 441 = √9 + 81 = √160 = √490 = √90 = 4√10 = 7√10 = 3√10 . . . Are A, B, and C collinear? If yes, the two small distances add to the large distance. 4√10 + 3√10 = 7√10 Since BC is the longest distance, B and C are the endpoints. The points are collinear!! Thus, point A must be in the middle.

Pick one, #1 or #2!! 12. Show that the triangle with vertices P(5, 0), E(–1, –2), and T(3, 6) is isosceles. PE = √(5+1)2 + (0+2)2 ET = √(-1 – 3)2 + (-2 – 6)2 PT = √(5 – 3)2 + (0 – 6)2 = √36 + 4 = √16 + 64 = √4 + 36 = √40 = √80 = √40 = 2√10 = 4√5 = 2√10 The triangle is isosceles because two of its sides are the same length! 13. Quadrilateral GEMA has vertices G(3, 8), E(8, –3), M(–2, –5), and A(–5, 2). Show that its diagonals are congruent. GM = √(3+2)2 + (8+5)2 EA = √(8+5)2 + (-3 – 2)2 = √25 + 169 = √169 + 25 = √194 = √194 No need to simpilfy… The diagonals are congruent!

This is the circle formula to remember!! An equation of the circle with center (a, b) and radius r is: This is the circle formula to remember!! Why is the circle formula in the section about the distance formula? If you rearrange the formula above it reads… Now that we know the distance formula, this represents all points that are “r distance” away from a point (a, b). This is precisely a circle!!!

How could this be a circle? An equation of the circle with center (a, b) and radius r is: How could this be a circle? Let’s analyze (x – 0)2 + (y – 0)2 = 81 to see if it really is a circle!!

Center: (-3, 5) Radius = 2 x2 + y2 = 81 (x + 3)2 + (y + 8)2 = 36 Example 4 Find the center and radius of the circle with the equation: Center: (-3, 5) Radius = 2 Write an equation of the circle that has the given center and radius. 14. C(0, 0); r = 9 15. C(-3, -8); r = 6 16. C(1, -2); r = x2 + y2 = 81 (x + 3)2 + (y + 8)2 = 36 (x – 1)2 + (y + 2)2 = 3

. . . . Center: (2, -4) Center: (-7, -3) Radius = 3 Radius = 5 Find the center and radius of each circle. Sketch the graph. 17. 18. Center: (2, -4) Center: (-7, -3) Radius = 3 Radius = 5 . . 19. 20. . . Center: (1, 2) Center: (0, 1) Radius = 3/4 Radius = 7

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